Relation of DWNs to FDNs

Digital Waveguide Networks

We have just seen how the FDN can be seen as a simple DWN having a not-necessarily physical scattering matrix. In order to provide an easy control over the decay, the scattering matrix has to satisfy the condition of losslessness (27). A finite-precision implementation of the FDN might incur in limit cycles or overflow oscillations, due to departures from the infinite-precision lossless prototype. Departures can be of two kinds: the finite-precision scattering matrix does not satisfy the lossless condition (27), or the round-off noise in the matrix by vector multiplication introduces signal amplitude modifications. By assuming that the scattering matrix satisfies (27) in a large extent even in finite precision, it is possible to apply the arguments used in [27,28,26,6] for the DWNs, in order to avoid limit cycles or overflow oscillations. If the matrix by vector multiplication is performed in the straightforward way as a collection of inner products, and the matrix coefficients have the same bits of precision as the signals, it is just sufficient to perform these order- inner products in the extended precision of bits, and apply a passive truncation scheme on the output signal. In two's complement arithmetic, a simple passive truncation scheme is the following:

- If the N-1 most significant bits are not equal, replace the output value by the maximum-magnitude number in -bit two's complement having the correct sign (saturation).
- Discard the least significant bits and add to the result if it is negative.

Circulant Feedback Delay Networks

Relation of DWNs to FDNs

Digital Waveguide Networks

``Circulant and Elliptic Feedback Delay Networks for Artificial Reverberation'', by Davide Rocchesso and Julius O. Smith III, preprint of version in

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